How to check if a known object is intersecting with an object using python
Use some kind of quick bounding box culling like this (javascript demo on collision detection by mozilla, but should be easy enough to adapt for your case) to eliminate objects that cannot overlap. Beware that Blender objects' provided bounding box is in object space, so may not be axis aligned in world space. You will need to convert it to world space and then generate a new bounding box of that bounding box that is axis aligned.
Then build bvh trees for each mesh that may intersect, and test, re-using the tree for your test object.